Horizon universality and anomalous conductivities
aa r X i v : . [ h e p - t h ] O c t ICCUB-14-060
Horizon universality and anomalousconductivities
Umut G¨ursoy and Javier Tarr´ıo Institute for Theoretical Physics and Spinoza Institute, Utrecht University3508 TD Utrecht, The Netherlands Departament de F´ısica Fonamental and Institut de Ci`encies del Cosmos,Universitat de Barcelona, Mart´ı i Franqu`es 1, ES-08028, Barcelona, Spain.e-mails: [email protected], [email protected]
Abstract
We show that the value of chiral conductivities associated with anoma-lous transport is universal in a general class of strongly coupled quantumfield theories. Our result applies to theories with no dynamical gluon fieldsand admitting a gravitational holographic dual in the large N limit. On thegravity side the result follows from near horizon universality of the fluctua-tion equations, similar to the holographic calculation of the shear viscosity.
Introduction
Anomalous transport in quantum field theories with chiral fermions has enjoyed a re-newal of interest since the recent discovery of the Chiral Magnetic Effect (CME) [1, 2].In short, the CME refers to generation of a macroscopic electric current as a result ofthe axial anomaly in the presence of an external magnetic field ~B~J = σ VV ~B . (1)Here the “chiral magnetic conductivity” σ VV is proportional to the anomaly coefficients.The main motivation to study the CME in the context of particle physics stems fromits possible realization in the Heavy Ion Collision experiments. Indeed, in an off-central collision of heavy ions at RHIC and LHC, huge magnetic fields are expected tobe generated by the “spectator” ions that do not participate in the formation of theQuark-Gluon Plasma (QGP) [3]. Then one can theoretically demonstrate [1, 2] thatthe chiral anomaly in QCD with electromagnetic and gluon contributions gives rise tothe CME in the off-central heavy ion collisions.Presence of such anomaly-induced electric and chiral currents in the QGP mighthave imprints on the spectra of charged hadrons observed in the heavy ion collisions.Such experimental evidence is still controversial at present [4]. It is important to notethat anomalous transport may also be observable in certain condensed matter systems,such as the —so far theoretical— constructions called the Weyl semi-metals, that canbe viewed as strongly coupled electron-hole plasmas in 3 spatial dimensions with thesingle-particle excitations being chiral fermions [5, 6, 7].In this paper we address the question whether the anomalous conductivities receiveradiative corrections. We address this question in the holographic setting with as muchgenerality as possible. The axial current is known to enjoy both electromagnetic andQCD quantum anomalies that lead to the anomaly equation, ∂ µ J µ = a F A ∧ F A + a F V ∧ F V + a Tr G ∧ G , (2)where F A,V are the field strengths of background axial and vector gauge fields, G isfield strength of the gluon field and a i are anomaly coefficients that are well-known tobe one-loop exact [8].In the absence of perpendicular external (axio-)electric and (axio-)magnetic fieldsthe first two terms in (2) vanish. The last term in (2) then is the main source ofanomaly induced chiral imbalance. One can then effectively take into account suchanomaly generating glue transitions by introducing an axial chemical potential µ .Then, one can show by various different methods [1, 2], that the presence of theelectromagnetic anomaly a = 0 in the presence of an external magnetic field ~B leadsto generation of an electric current as in (1), with σ VV = e π µ . (3) Even though there is no background axial gauge fields in nature, here we include them for gener-ality. In QCD at finite temperature the most dominating such process is argued to be the the sphalerondecays [9] σ VV isthe chiral magnetic conductivity . There is a similar effect, namely generation of a chiralcurrent J in response to a magnetic field that is J = σ AV B . This effect is called the chiral separation effect .A great deal of theoretical research in anomalous transport is focused on whetherthe value of the chiral magnetic conductivity above (and similarly other anomalousconductivities) receives radiative corrections or not. There exist a variety of argumentsin favor of —at least perturbative— non-renormalization [10, 2, 11, 12, 13, 14, 15],mainly due to the fact that the anomaly coefficients are one-loop exact [8]. The situationhowever is subtle and to make a clear statement about non-renormalization one has todistinguish the two types of anomalies that we will call type I and type II [16, 17].The former type refers to anomalies that vanish when the external fields are turnedoff (i.e. they are ’t Hooft anomalies), such as the first and the second terms in (2),whereas the type II anomalies refer to mixed gauge-global anomalies such as the gluoniccontribution in (2) whose presence does not depend on external fields. In the lattercase, one does expect radiative corrections generically [16].In the absence of type II anomalies, on the other hand, there exists both direct andindirect methods establishing non-renormalization. Firstly, one can show absence ofperturbative corrections directly in field theory using the axial and vector Ward identi-ties and some recently proven non-renormalization theorems [18], see for example [19].Absence of non-perturbative renormalization can also be established by two indirectmethods. Firstly, assuming that a hydrodynamic description of the system is at hand,demanding a positive definite divergence of the entropy current determines σ VV to beexactly as in (1), as shown in [20] (see also [21]). Secondly, [22] established a Euclideaneffective field theory for anomalous transport, whose consistency again requires fixingthe value of σ VV as in (1).In this note we address the question of whether the value of anomalous conductivi-ties such as σ VV in (1) are exact in strongly coupled quantum field theories that admita gravitational dual description a la AdS/CFT [23, 24, 25]. The study of anomaloustransport via the holographic correspondence played a major role in the development ofthe subject from early on [26, 27]. In holography one introduces anomalous currents byconsidering bulk gauge fields in the presence of bulk Chern-Simons terms [25]. One canthen calculate the anomalous conductivities using the Kubo’s formulae by calculatingthe retarded Green’s functions following the standard prescription of the holographiccorrespondence. The case of conformal plasma of N = 4 super Yang-Mills in the large N c limit was first to be considered in the holographic description, in a series of papersby Landsteiner et al. [28, 29, 30, 31]. By comparison of the holographic and weak cou-pling results, these authors concluded that the chiral magnetic conductivity receivesno corrections at all. However, this is a very special theory, and one is immediatelyprompted to analyze the situation in a more general class of theories, in particulartheories with a mass gap and running gauge coupling. Such a study was undertakenvery recently by one of the authors together with A. Jansen in [32]. In that paperthe anomalous conductivities were calculated in a holographic setting that is dual to anon-conformal theory that exhibits a confinement-deconfinement transition. One finds2hat the value in (1) non-trivially depends on the parameters of the gravitational back-ground, hence it seems that the universal value in (1) no longer holds. However, whenthe result is expressed in terms of physical quantities such as the chemical potential µ and temperature T , one again finds that the anomalous conductivities attain theiruniversal values. In particular the chiral magnetic conductivity is again precisely givenby (1) [32].The non-trivial result obtained in [32] prompted us to seek for a generic background-independent mechanism to explain the universality of anomalous conductivities in theholographic setting. In a sense, in this paper we seek for the holographic analog ofthe non-renormalization theorems in field theory , that we summarized above. Suchuniversal values for the transport coefficients would typically result from the universalnear-horizon behavior of bulk fluctuations in black-hole backgrounds. The most famousexample of such behavior is the universal value of the shear viscosity to entropy densityratio η/s = 1 / π in gravitational theories quadratic in derivatives [33, 34]. This robustresult can indeed be explained by the background-independence of metric fluctuationsnear the horizon, see for example [35].In case of the anomalous conductivities, such direct proofs of universality prove dif-ficult because, unlike the case of the shear viscosity or electric conductivity, calculationof anomalous conductivities in holography involves mixing of bulk gauge and metricfluctuations. In a way, in order to establish universality one has to diagonalize thesefluctuations which turns out to be an onerous task. A simpler case was studied in[36], where the authors considered gravity theories with no scalars and looked at theholographic flow of the chiral condutivities, i.e. their dependence on the radial coor-dinate. Particularly, in the case of the AdS-Reissner-Nordstr¨om blackhole, they foundno holographic flow except the scale dependence of the chemical potentials.In this paper, we address the calculation in a general class of two-derivative gravitymodels in an alternative way, namely by including the source, i.e. , the axial or the vec-tor magnetic field, in the fluctuations themselves. This method was already introducedby Donos and Gauntlett in a different context where the authors study the thermo-electric properties of holographic plasmas [37]. Employing this method we prove theuniversality of anomalous conductivities such as the chiral magnetic and chiral separa-tion conductivities for a quite general action. This result establishes the holographicanalog of the non-renormalization theorems that are found on the field theory side.The paper is organized as follows.In section 2, we introduce the general holographic setting that we employ in thispaper. Here we also fix the coefficients of the Chern-Simons terms by matching theanomaly equation on the field theory side.In section 3 we introduce our ansatz for the fluctuation equations in order to cal-culate the anomalous conductivities. We study the near boundary and near horizonbehavior of these fluctuations and show that the regularity of metric fluctuations nearthe horizon require vanishing of metric fluctuations there. In this section we also deriveexpressions for the conserved fluxes that correspond to these fluctuations.In section 4 we finally evaluate the anomalous conductivities associated with back-ground vector and axial sources and demonstrate universality in their values.The final section discusses our results and the methods and present an outlook for3urther research. The model we work with in this paper is given by the following action S = 116 π G Z " R ∗ −
12 d φ ∧ ∗ d φ − Ψ( φ )2 d χ ∧ ∗ d χ − V ( φ ) ∗ − Z A ( φ )2 F A ∧ ∗ F A − Z V ( φ )2 F V ∧ ∗ F V + κ A ∧ (cid:0) F A ∧ F A + 3 g F V ∧ F V (cid:1) , (4)with F A = d A the axial field and F V = d V the vector one. The dilaton scalar field, φ ,and the axion one, χ , will not play an explicit rˆole in the solution of the fluctuations,even when we will assume that they are not-trivial in the background. The main effectof the dilaton in this study comes through the dilatonic couplings Z A,V . Here we assumethat the one-forms A and V are normalized in such a way thatlim r →∞ Z A ( φ ) = lim r →∞ Z V ( φ ) = 1 , (5)with r → ∞ corresponding to the boundary, where the metric is asymptotically AdS .The normalization (5) can always be attained with a redefinition of the factors κ and g in the Chern-Simons (CS) terms.The choice of CS terms in (4) corresponds to including a Bardeen counter-term inthe boundary action [28] whose presence is required for an anomaly-free vector current.Indeed if we make a gauge transformation V → V + d ζ V in (4) we obtain ∂ µ J µ = 0 forthe vector current that is dual to the bulk gauge-field V µ .The coefficients κ and g are not arbitrary, and their value can be found matchingto the gauge anomaly of one left-handed and one right-handed fermion. With a gaugetransformation δ ζ A for the axial U(1) field A we obtain δ ζ A S = κ π G (cid:0) F A ∧ F A + 3 g F V ∧ F V (cid:1) = − ∂ µ J µ . (6)On the other hand, in the presence of the Bardeen counter-term on the boundary theory,the correct anomaly equation for the axial current reads, ∂ µ J µ = 112 π (cid:0) F V ∧ F V + F A ∧ F A (cid:1) . (7)Matching (6) with (7) we determine κ = − G N c π , g = 1 . (8)From now on we set g = 1. 4he equations of motion from the variation of the action (4) read for the scalarsd (Ψ ∗ d χ ) = 0 , (9)d ∗ d φ = ∂ φ V ∗ ∂ φ Z A F A ∧ ∗ F A + ∂ φ Z V F V ∧ ∗ F V + ∂ φ Ψ2 d χ ∧ ∗ d χ , (10)for the gauge fields d (cid:0) Z A ∗ F A − κ A ∧ F A − κ V ∧ F V (cid:1) = 0 , (11)d (cid:0) Z V ∗ F V − κ A ∧ F V (cid:1) = 0 , (12)and finally for the metric R µν = 12 ∂ µ φ∂ ν φ + Ψ2 ∂ µ χ∂ ν χ + V g µν + Z A (cid:18) F Aµρ F Aν ρ − g µν F Aρσ F A,ρσ (cid:19) + Z V (cid:18) F Vµρ F Vν ρ − g µν F Vρσ F V,ρσ (cid:19) . (13)We will assume a static, translation- and rotation-invariant background, given bythe following configurationd s = − g tt ( r )d t + g xx ( r )d ~x + g rr ( r )d r , (14) A = A t ( r )d t , V = V t ( r )d t , φ ( r ) , χ ( r ) . (15)Notice that g tt is a positive-definite function, and that AdS-RN with two charges fallsinto this general ansatz for the specific values Z A = Z V = 1, V = − χ = φ = 0. Ifwe further require V t = 0 we recover the case studied in section 3.2 of [36].In the UV, r → ∞ , we will require that the solution becomes asymptotically AdS g tt ∼ r + · · · , g xx ∼ r + · · · , g rr ∼ r − + · · · , (16)with the remaining functions going to constants: A t ∼ A ∞ t + · · · , etc. The dots indicatesubleading terms.For the IR of the theory we require the existence of a non-extremal horizon, suchthat near the horizon, r = r h , we have g tt ∼ t h ( r − r h ) + · · · , g xx ∼ x h + · · · , g rr ∼ ρ h r − r h + · · · , (17)with the remaining functions going to constants: A t ∼ A ht + · · · , etc.Before analysing the fluctuations of the system it is useful to express the temporalcomponents of the gauge fields, A t and V t , in terms of constants of motion. To this endlet us define first J µν = −√− g Z A ( φ ) F A,µν + κ ǫ µναρσ (cid:0) A α F Aρσ + a α F Vρσ (cid:1) , (18) J µν = −√− g Z V ( φ ) F V,µν + κ ǫ µναρσ A α F Vρσ , (19)5rom where the equations of motion (11) and (12) read simply ∂ µ J µν = 0 , ∂ µ J µν = 0 . (20)For the background (14) the only non-vanishing components are J rt ,bg = s g xx g tt g rr Z A ( φ ) A ′ t , J rtbg = s g xx g tt g rr Z V ( φ ) V ′ t , (21)which from the equations of motion are constants J rt ,bg = Q , J rtbg = Q . (22)
In this section we give the configuration of fluctuations that will give rise to the anoma-lous transport coefficients. As shown for example in [28], it is consistent to restrict thestudy to the following set of fluctuations δA = δA x ( y, r )d x + δA z ( y, r )d z , (23) δV = δV x ( y, r )d x + δV z ( y, r )d z , (24) δ d s = 2 δg tx ( y, r )d t d x + 2 δg tz ( y, r )d t d z . (25)These fluctuations correspond to the vectorial sector preserving rotations in the x – z plane, which is the reason why they do not couple to fluctuations of other componentsof the metric or the gauge fields, nor to the scalars.In this moment we will employ the method in [37] to express the sources explicitelyin the fluctuations. We will turn these on for the gauge fields only in the following way δA x ( y, r ) = − B z y + α x ( r ) , δA z ( y, r ) = B x y + α z ( r ) , (26) δV x ( y, r ) = − B z y + β x ( r ) , δV z ( y, r ) = B x y + β z ( r ) , (27) δg tx ( y, r ) = g xx γ x ( r ) , δg tz ( y, r ) = g xx γ z ( r ) , (28)with the magnetic field sources B a and B a , with a = { x, z } . Now, we want the α a and γ a to correspond to normalizable deformations of the fields, thus describing theresponse to the B (5) a sources. Since the gauge fields and the graviton are massless,the holographic correspondence implies that we must have near the UV the followingleading behavior: α a ∼ r − , β a ∼ r − and γ a ∼ r − .To check this asymptotic behavior let us begin analyzing the equations of motionfor the gauge fields fluctuations. With the definitions (18) and (19) these equations are ∂ r δJ r a + ∂ y δJ y a = 0 , ∂ r δJ r a + ∂ y δJ y a = 0 , (29)with δJ µ a (5) the part of (19) ((18)) linear in fluctuations with our background ansatz(14). 6t turns out that the derivatives with respect to the spatial coordinate y can beexpressed as radial derivatives ∂ y δJ y a = − (cid:0) κ A ′ t B b + κ V ′ t B b (cid:1) δ ab , ∂ y δJ y a = − κ V ′ t B b δ ab , (30)with δ ab Kronecker’s delta. Therefore the equations of motion for the fluctuations ofthe gauge fields read ∂ r ˜ J a = ∂ r ˜ J a = 0 , (31)where we have defined˜ J a ≡ − (cid:18) κ A t B b + 2 κ V t B b + Q γ b + r g tt g xx g rr Z A ( φ ) α ′ b (cid:19) δ ab , (32)˜ J a ≡ − (cid:18) κ A t B b + 2 κ V t B b + Q γ b + r g tt g xx g rr Z V ( φ ) β ′ b (cid:19) δ ab . (33)The ˜ J a (5) quantities are conserved on-shell along the radial direction. In particularthey help determining the leading behavior of the α a and β a fluctuations in the UV.Provided γ a ∼ r − we indeed get α a ( r ) ≃ ˜ J b δ ab + 2 κ ( A ∞ t B a + V ∞ t B a )2 r + · · · , (34) β a ( r ) ≃ ˜ J b δ ab + 2 κ ( A ∞ t B a + V ∞ t B a )2 r + · · · , (35)at large radius, and notice the shift with respect to the conserved quantities ˜ J a in thenumerator.We must now prove that consistently γ a ∼ r − near the boundary. This is straight-forward to see once we realize that the equations of motion for the fluctuations γ a aresimply ∂ r ˜ K a = 0 , (36)where we have used the background equations of motion and defined two more constantsof motion ˜ K a ≡ s g xx g tt g rr γ ′ a + Q α a + Q β a . (37)Evaluating ˜ K a at the UV and using the boundary behavior of the backgroundsolutions we find at large radius γ µ ( r ) ≃ − ˜ K µ r + · · · , (38)and we recover from the fluctuation equations the expected behavior for the functions.We study now the behavior of the fields near the horizon. With this in mind letus first quote the result for the scalar of curvature with the fluctuations of the metric(26): R = R bg + S a γ a + S ab γ a γ b g tt , (39)7here S a and S ab are some radial functions regular at the horizon, and R bg the Ricciscalar of the background solution. From this expression we see that near the horizonone has a curvature singularity unless the γ a vanish there.Plugging this behavior in (32), (33) and (37) the leading behavior of α a compatiblewith the equations is that the fluctuations of the gauge fields go to constants at thehorizon.With these two results in hand we can determine the value of the constants ofmotion ˜ J a (5) straightforwardly at the horizon˜ J a = − (cid:0) κ A ht B b + κ V ht B b (cid:1) δ ab , (40)˜ J a = − κ (cid:0) A ht B b + V ht B b (cid:1) δ ab . (41)Once we have determined the behavior of the fields near the horizon and near theboundary we have specified completely the solutions to α a , β a and γ a . In the nextsection we will see how this is enough to build the one-point functions associated to thefluctuations we have been considering. Once equipped with the expressions (40) and (41) we can write the one-point functionscorresponding to the expected value of the axial and vector currents. This is obtainedfrom the normalizable mode of the α a (in equation (34)) and β a (in equation (35))fluctuations appropriately normalized. From action (4) the final result reads h J a i = − κ π G (cid:2) ( A ∞ t − A ht ) B b + ( V ∞ t − V ht ) B b (cid:3) δ ab = − κ π G (cid:0) µ B b + µ B b (cid:1) δ ab , (42) h J a i = − κ π G (cid:2) ( A ∞ t − A ht ) B b + ( V ∞ t − V ht ) B b (cid:3) δ ab = − κ π G (cid:0) µ B b + µ B b (cid:1) δ ab , (43)with A ∞ t − A ht = µ and V ∞ t − V ht = µ the associated chemical potentials. Plugging inthe value for κ in (8) we get for the one-point functions h J a i = N c π (cid:0) µ B b + µ B b (cid:1) δ ab , (44) h J a i = N c π (cid:0) µ B b + µ B b (cid:1) δ ab . (45)From these expressions we define the chiral conductivities as the derivatives of the one-point function with respect to each of the magnetic fields. Since the dependence onthese is linear we can simply write h J a i = σ VV B a + σ VA B a , (46) h J a i = σ AA B a + σ AV B a , (47) Strictly speaking one should choose the gauge A ∞ t = V ∞ t = 0 to avoid complications in thecalculation by a redefinition of (42), see [38]. We discuss this point further in the Discussion sectionbelow. σ VV = σ AA = N c π µ , σ AV = σ VA = N c π µ . (48)In particular, upon setting the axial magnetic field to zero, we obtain the desired chiralmagnetic effect h J ν i = σ VV B ν = N c π µ B ν , (49)and chiral separation effect h J ν i = σ AV B ν = N c π µ B ν . (50)We emphasize that our results (49) and (50) follows directly from the horizon univer-sality of the fluctuations and, as such, they are valid in a generic class of theories wherethe gravitational description solves the generic action (4). We demonstrated that the anomalous conductivity describing the chiral magnetic ef-fect acquires an universal value for a generic holographic model, independently of thedetails of the background solution on which this effect is calculated. As we discussedin the Introduction, this fact was already shown to hold on the field theory side with avariety of different methods including the Ward identities and new non-renormalizationtheorems, hydrodynamics and effective field theories. Our calculation provides yet an-other, independent demonstration of this non-renormalization and fills in the gap onthe dual gravitational side. It is reassuring to find that there exist a holographic analogto this field theory non-renormalization theorem, and it indeed follows from horizonuniversality, as one would have expected.The only requirement that we impose in our construction, besides the general formof the action, is a rather physical one: that the curvature scalar is not divergent onthe horizon. This allowed us to express the conserved quantities (32), (33) and (37)in terms of horizon quantities, which recombined in a neat way with UV data in thesolution for the fluctuations near the boundary to produce the chemical potentials ofthe theory in the final result.We left out the calculation of the chiral vortical effect in our analysis. This can bestudied by adding an extra piece to the action (4) S → S + λ πG Z A ∧ tr ( R ∧ R ) , (51)with R µν = R µνρσ d x ρ ∧ d x σ the curvature tensor. Additionally one needs to add a newcounterterm to the action to make the variational problem well posed. The effect of thisnew piece of the action in the equations of motion is to add an extra piece that behavesas λ tr( R ∧ R ), in (11), and a new one in (13) as well. For the fluctuations consideredin this paper the former term vanishes, impliying that the result for the anomalousconductivity cannot change due to the presence of the new term in the action. We plan9o present the calculation for all anomalous transport coefficients including the chiralvortical effect in the future, thus providing the generalization for a gravity theory withscalar matter of the results in section 4 of [36].We should also comment on a technical issue that was alluded to in the footnote onthe previous page. Our result should of course be independent of the choice of gauge forthe bulk gauge fields. However, there is a subtlety [38] arising from different methodsto realize the chemical potentials µ and µ on gravity side. In the first method, that weemployed here, we choose a vanishing value for the boundary values of the gauge fields.In this method the calculation goes straightforwardly with the definition (42). Onemay also choose the gauge fields to vanish on the horizon and asymptote to finite valueson the boundary. In this case however there is another contribution to the boundarychiral current (42) that arises from the finite Chern-Simons current that is added to the consistent current to obtain a covariant current. To establish gauge equivalence withthe previous method one has to include a spurious boundary axion [38]. We choose towork with the first method, i.e. the formalism B in the nomenclature of [38].As is common with general calculations showing a robust result, the setup we haveconsidered in this paper may help identify ways to model setups where the chiral mag-netic conductivity differs from the universal result found here. One such possibility isto try to prevent the equation of motion for the axial gauge field fluctuation to have aconstant of motion, (31). This follows for example from a St¨uckelberg type of action,in which the kinetic term for the axion in (4) now reads(16 πG ) S χχ = − Z Ψ( φ )2 (d χ − m A A ) ∧ ∗ (d χ − m A A ) , (52)with m A the mass of the axial field. Indeed this mechanism was proposed as a holo-graphic dual to anomalous theories with type II anomalies —that is, with a gluoniccontribution to the anomaly equation ( a = 0 in equation (2))— in [32], see also [39].The bulk axion is dual to a theta-term in the field theory action. Coupling χ to theaxial field induces a non-trivial value for the h Tr G ∧ G i expectation value in (2). Theanomaly term associated with this expectation value, a , then induces a modificationin the result for the anomalous transport coefficients [32, 39]. In holographic construc-tions of QCD m A is usually proportional to the number of flavors in the theory, andone needs a calculation in the Veneziano limit to see the effect of this term [40].There are possible various extensions of our work. First of all, as already mentionedabove, the extension of the holographic non-renormalization to the case of chiral vor-tical conductivity would be very interesting. Secondly, one may wonder whether ouruniversal result for the chiral conductivities survive the higher derivative corrections ingravity, or not. It is well-known that one generates corrections to the shear viscosity inpresence of higher derivative corrections [41], but the notion of horizon universality con-tinues to hold. In the case of anomalous transport we also expect horizon universalityto determine the values of conductivities exactly, also in presence of higher derivativeterms. Whether the actual value changes or not remains to be seen. In this contextone should note a physical distinction between viscosity and anomalous conductivities,namely the former is dissipative and the latter is not. It is conceivable therefore thatthe result we found here may be robust against higher derivative corrections. Finally,10ne may wonder if the kind of universality we find here extends to finite frequency andmomenta. It would be also very interesting to predict such universal behavior in themomenta dependence of the anomalous conductivities in holography. Acknowledgements
We thank Francisco Pena-Benitez, Aron Jansen and especially Aristomenis Donos forinteresting discussions and useful remarks. JT is grateful to the Mainz Institute forTheoretical Physics (MITP) for its hospitality and partial support during the initialstage of this work.This work is part of the D-ITP consortium, a program of the Netherlands Organisa-tion for Scientific Research (NWO) that is funded by the Dutch Ministry of Education,Culture and Science (OCW).JT is supported by grants 2014-SGR-1474, MEC FPA2010-20807-C02-01, MECFPA2010-20807-C02-02, ERC Starting Grant HoloLHC-306605 and by the Juan de laCierva program of the Spanish Ministry of Economy.
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